J-self-adjointness, Krein spaces and related physics

J-self-adjointness, Krein spaces and related physics

Günther, U.

As brief introduction, a few basic aspects of the theory of J-self-adjoint operators and matrices are sketched as well as their natural relation to Krein spaces --- Hilbert spaces with indefinite metric structures. The mathematical facts are then illustrated by some recent results from PT quantum mechanics:
  • the highly nontrivial spectral behavior of a quantum mechanical Hamiltonian with x2(ix)ν potential in the sector of spontaneously broken PT-symmetry, the form invariant resolution of the spectral singularity in the limit ν → -1 and a hidden UV-IR duality
  • the unfolding of higher-order branch-points (exceptional points) in PT-symmetric Bose-Hubbard models and the relevance of the underlying Hessenberg type of perturbation matrices
  • the Naimark dilation of the PT-symmetric quantum brachistochrone solution with the unexpected physical result of inducing ultra-fast (wormhole-like) evolution regimes via fine-tuned entanglement
The talk concludes with briefly indicating further applications and possible future developments.

Keywords: J-self-adjoint operators; Krein space; PT quantum mechanics; spontaneously broken symmetry; UV-IR duality; Bose-Hubbard model; Bose-Einstein condensate; exceptional point; branch point; brachistochrone; entangled states; ultra-fast evolution; wormhole-like behavior

  • Invited lecture (Conferences)
    Editorial Board Meeting; Journal of Physics A: Mathematical and Theoretical, 23.-24.04.2009, London, UK

Permalink: https://www.hzdr.de/publications/Publ-12510